Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

三元双正交小波滤波器的刻画

研究由三元双正交插值尺度函数构造对应的双正交小波滤波器的矩阵扩充问题.当给定的一对三元双正交尺度函数中有一个为插值函数时,利用提升思想与矩阵多相分解方法,给出一类三元双正交小波滤波器的显示构造公式和一个计算实例.讨论了三元双正交小波包的的性质.     

Construction of Biorthogonal Discrete Wavelet Transforms Using Interpolatory Splines

We present a new family of biorthogonal wavelet and wavelet packet transforms for discrete periodic signals and a related library of biorthogonal periodic symmetric waveforms. The construction is based on the superconvergence property of the interpolatory polynomial splines of even degrees. The construction of the transforms is performed in a “lifting” manner that allows more efficient implementation and provides tools for custom design of the filters and wavelets. As is common in lifting schemes, the computations can be carried out “in place” and the inverse transform is performed in a reverse order. The difference with the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Our algorithm allows a stable construction of filters with many vanishing moments. The computational complexity of the algorithm is comparable with the complexity of the standard wavelet transform. Our scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. In addition, these filters yield perfect frequency resolution.     

Wavelet Filtering for Filtered Backprojection in Computed Tomography

In this paper, we present a characterization of MRA biorthogonal wavelet filters with full frequency supports. Based on this characterization, it is established that wavelet ramp filters are biorthogonal wavelets if the original wavelets are sufficiently regular. An efficient subband coding algorithm is developed for wavelet filtering in filtered backprojection, which is the most popular method in computed tomography (CT). Computer simulation suggests that this wavelet filtering process is a useful tool for improving image quality and reducing computational time in local CT reconstruction.     

M进制双正交对称小波的频率优化设计

§ 1　IntroductionM-band wavelets are used in a lotof applications such asspeech coding,image analysisand image coding[1 ,2 ] .When applied to the low-rate subband image coding,the symmetricextension method[3] has been shown to outperform the circular convolution method andyield both objective and subjective quality improvementat image boundaries.The symmet-ric extension method requires linear-phase scaling filters and wavelet filters.Two-bandbiorthogonal symmetric waveletbases have been const…     

Biorthogonal bases of compactly supported wavelets

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.     

Biorthogonal M-band filter construction using the lifting scheme

The lifting scheme has been proposed as a new idea for the construction of 2-band compactly supported wavelets with compactly-supported duals. The basic idea behind the lifting scheme is that it provides a simple relationship between all multiresolution analyses sharing the same scaling function. It is therefore possible to obtain custom-designed compactly supported wavelets with required regularity, vanishing moments, shape, etc. In this work, we generalize the lifting scheme for the construction of compactly-supported biorthogonal M-band filters. As in the previous case, we used the flexibility of the scheme to exploit the degree of freedom left after satisfying the perfect-reconstruction conditions in order to obtain finite filters with some interesting properties, such as vanishing moments, symmetry, shape, etc., or that satisfy certain optimality requests required for particular applications. Moreover, for these lifted biorthogonal M-band filters, we give an analysis-synthesis algorithm which is more efficient than the standard algorithm realized with filters with similar compression capabilities. This revised version was published online in June 2006 with corrections to the Cover Date.     

Frequency optimization approach for designing <Emphasis Type="Italic">M</Emphasis>-band biorthogonal symmetric wavelets

In this paper, a new method is presented for designing M-band biorthogonal symmetric wavelets. The design problem of biorthogonal linear-phase scaling filters and wavelet filters as a quadratic programming problem with the linear constraints is formulated. The closed-form solution is given and a design example is presented.     

N DIMENSIONAL FINITE WAVELET FILTERS

In this paper, a large class of n dimensional orthogonal and biorthognal wavelet filters(lowpass and highpass) are presented in explicit expression. We also characterize orthogonal filters with linear phase in this case. Some examples are also given, including non separable orhogonal and biorthogonal filters with linear phase.     

Biorthogonal radial multiresolution in dimension three

In this paper, we present the definition and the relative theorems of the biorthogonal radial multiresolution in dimension three. Unlike the orthogonal case, there exist real-valued dual radial scaling functions with compact support in the biorthogonal case. The associated Mallat algorithm can be simply performed in terms of classical biorthogonal filters.     

利用提升格式构造稳定的对偶小波

近年来 ,产生了一种称为“提升格式”的新的小波构造方法 [7,8,9] ,它从一个较简单的多尺度分析 (MRA)出发 ,利用尺度函数相同的多尺度分析之间的相互关系 ,逐步地得到所需性质的多尺度分析 .本文仅考虑双正交滤波的提升格式 .当选定一初始双正交滤波后 ,利用提升格式构造的双正交滤波仍是双正交的 ,而这双正交滤波能否生成双正交小波 Riesz基即稳定的对偶小波 ?更进一步 ,如何从一些较为简单的不能生成双正交小波 Riesz基的双正交滤波出发 ,利用提升格式构造出具有 Riesz基性质的双正交滤波 ?这在目前有关提升格式的文章中没作回答 .本…     

利用MATLAB符号计算求解双正交小波滤波器系数

给出了用MATLAB符号计算求解双正交小波的滤波器系数(有理数)的程序,并更正了某些文献中的滤波器系数中的错误.     

Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

The lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. Here it is extended to the construction of multiwavelets. It is shown that any set of compactly supported biorthogonal multiwavelets can be obtained from the Lazy matrix filters with a finite number of lifting steps. As an illustration of the general theory, compactly supported biorthogonal multiwavelets with optimum time–frequency resolution are constructed. In addition, experimental results of applying these multiwavelets to image compression are presented.     

Design of Interpolating Biorthogonal Multiwavelet Systems with Compact Support

In this paper we characterize all totally interpolating biorthogonal finite impulse response (FIR) multifilter banks of multiplicity two, and provide a design framework for corresponding compactly supported multiwavelet systems with high approximation order. In these systems, each component of the analysis and synthesis portions possesses the interpolating property. The design framework is based on scalar filter banks, and examples with approximation order two and three are provided. We show that our multiwavelet systems preserve almost all of the desirable properties of the generalized interpolating scalar wavelet systems, including the dyadic-rational nature of the filter coefficients, equality of the flatness degree of the low-pass filters and the approximation order of the corresponding functions, and equality between the uniform samples of a signal and its projection coefficients for a given scale. This last property allows us to avoid the cumbersome prefiltering associated with standard multiwavelet systems. We also show that there are no symmetric totally interpolating biorthogonal multifilter banks of multiplicity two. Finally, we point out that our design framework incorporates a simple relationship between the multiscaling functions and multiwavelets that substantially simplifies the implementation of the system.     

离散超小波变换下双正交小波谱分析

我们研究发现，在离散超小波变换下双正交小波谱是有界的．并且任何一个双正交小波变换的谱不可能分布在1附近的某个区间内，并给出了该区间的一个估计．     

Results on Biorthogonal Filter Banks

For a maximally decimated nonuniform filter bank, the perfect reconstruction (PR) property is equivalent to biorthogonality. We start from this result and derive a number of properties of PR filter banks. For example, no two integer decimators in a biorthogonal system can be coprime; moreover, if all analysis and synthesis filters have unit energy, then perfect reconstruction is equivalent to orthonormality. We also generalize the Nyquist and power complementary properties of orthonormal filter banks, for the biorthogonal case. We then show that whenever the decimation ratios are such that biorthogonality is possible with rational filters, it is, in particular, possible to obtain orthonormality with rational filters. This is done by developing an orthonormalization procedure. While reminiscent of the Gram–Schmidt approach, the procedure converges in a finite number of steps and furthermore preserves the filter-bank-like form of the basis functions. We also show how this technique can be applied for the decorrelation of subband signals. We will consider the problem of alias cancellation and obtain a generalization of a previously known necessary condition called compatibility.     

关于小波正则性的几个问题

本文给出了确定尺度函数正则指数的一个公式,证明了正交的四系数滤波中Daubechies小波正则性最优,并结合B-样条对应的双正交滤波,应用提升格式,增加对偶小波的正则性.     

Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities

We give a simple formula for the duals of the filters associated with bivariate box spline functions. We show how to construct bivariate non-separable compactly supported biorthogonal wavelets associated with box spline functions which have arbitrarily high regularities.     

Critical Exponent of Short Even Filters andBurt-Adelson Biorthogonal Wavelets

 We determine the critical exponent of all positive filters having an even residual of degree two and present an extension to the case of degree four. We apply these results to Burt-Adelson filters, thus determining the critical exponent of all the biorthogonal wavelets they generate. After this, we consider the problem of smoothing the dual wavelets by considering longer dual filters: we first create new wavelets by imposing an extra zero at π on the new filters and study their regularity by determining all the critical exponents. Then we release this condition on the filters and present the results of a numerical simulation intended to maximize the Sobolev regularity. (Received 10 February 2000; in revised form 16 May 2000)     

Multivariate compactly supported biorthogonal spline wavelets

We study biorthogonal bases of compactly supported wavelets constructed from box splines in ℝ ^N with any integer dilation factor. For a suitable class of box splines we write explicitly dual low-pass filters of arbitrarily high regularity and indicate how to construct the corresponding high-pass filters (primal and dual). Received: August 23, 2000; in final form: March 10, 2001?Published online: May 29, 2002